Math Insight

Single autonomous differential equation problems

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  1. For the differential equation \begin{align*} \diff{q}{t} = e^{-q^2-3q+1} \end{align*} find the equilibria and use the stability theorem to calculate their stability. Graph the solution for the initial condition $q(0)=0$.

  2. For the dynamical system \begin{align*} \diff{ u }{t} &= \left(- 6 u^{2} + 10 u\right) e^{- u}, \end{align*} find all equilibria and analytically determine their stability (i.e., use the stability theorem). Using this information, draw a phase line diagram with equilibria and vector field. (Be sure to indicate the stability of the equilibria.)

  3. Estimate the solution of the differential equation \begin{align*} \diff{ x }{t} &= 4 e^{- 1.7 x} - 7\\ x(0) &= 0.7, \end{align*} using the Forward Euler algorithm. Use a time step $\Delta t= 0.2$ to estimate $x(0.6)$.

  4. For the differential equation \begin{align*} \diff{z}{t} = h(z), \end{align*} the function $h(z)$ is graphed below. Autonomous differential equation example function 2
    1. Sketch the vector field illustrating the rate of change $\diff{z}{t}$.
    2. Find the equilibria and calculate their stability.
    3. Graph the solution $z(t)$
      1. for initial condition $z(0)=3.8$.
      2. for initial condition $z(0)=4.2$.

  5. Consider the dynamical system \begin{align*} \diff{u}{t} = u(2-u). \end{align*}
    1. Using any valid method, determine the equilibria of the dynamical system and their stability.
    2. Graph of the solution of the dynamical system with initial condition $u(0)=0.8$.
    3. Use the Forward Euler algorithm with time step $\Delta t=2$ to estimate the solution with initial condition $u(0)=0.8$. Take four time steps, estimating $u(2)$, $u(4)$, $u(6)$, and $u(8)$. Does the behavior of the solution estimated by Forward Euler match your previous results? If not, how is the behavior different?
    4. Use the Forward Euler algorithm with time step $\Delta t=1$ to estimate the solution with initial condition $u(0)=0.8$. Take four time steps, estimating $u(1)$, $u(2)$, $u(3)$ and $u(4)$. Does the behavior of the solution estimated by Forward Euler match your previous results? If not, how is the behavior different?
    5. Use the Forward Euler algorithm with time step $\Delta t=0.5$ to estimate the solution with initial condition $u(0)=0.8$. Take four time steps, estimating $u(1/2)$, $u(1)$, $u(3/2)$ and $u(2)$. Does the behavior of the solution estimated by Forward Euler match your previous results? If not, how is the behavior different?

  6. For the dynamical system $ \diff{ v }{t} = f(v,b),$ the function $f$ of $v$ depends on a parameter $b$, as shown in the graphs for $b=2, 8, 10, 12$, below. For values of $b$ in between those shown, $f$ changes smoothly, so its graph will be somewhere in between the snapshots shown. Sketch a bifurcation diagram with respect to the parameter $b$, for $2 \le b \le 12$. Use a solid line to indicate stable equilibria and a dashed line to indicate unstable equilibria. Identify any bifurcation points.

    $b=2$

    $b=10$

    $b=8$

    $b=12$

  7. Consider the differential equation \begin{align*} \diff{ s }{ t } &= -8.3 s. \end{align*}
    1. What is the general solution?
    2. What is the specific solution for the initial condition $s(0) = -1$?

  8. Consider the dynamical system \begin{align*} \diff{ u }{t} &= 2 \left(u + 2\right) \left(u + 1\right) \left(u - 7\right)\\ u(0) & = u_0, \end{align*} where $u_0$ is an initial condition. Sketch the graph of the function $f(u) = 2 \left(u + 2\right) \left(u + 1\right) \left(u - 7\right)$. Use the graph to sketch the solution $u(t)$ for each of the following initial conditions: $u_0 = -6.8$, $u_0 = -2$, $u_0 = -1.8$, $u_0 = -1$, $u_0 = 3.0$, $u_0 = 7$, and $u_0=9.4$.

  9. For the dynamical system $ u'(t) = h(u, \alpha)$, where the function $h$ of $u$ also depends on a parameter $\alpha$, a bifurcation diagram with respect to the parameter $\alpha$ is shown below. In this diagram, solid lines represent stable equilibria and dashed lines represent unstable equilibria.
    1. For the following three values of $\alpha$, determine the number of equilibria, their values, rounded to the nearest integer, and their stability. For each case, sketch the phase line, including equilibria and vector field. Use a solid circle for stable equilibria and an open circle for unstable equilibria.
      1. $\alpha= -5$
      2. $\alpha= -1$
      3. $\alpha= 7$
    2. Identify any bifurcation points.

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Elementary dynamical systems